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All Textbook Solutions for Single Variable Calculus: Early Transcendentals
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one. S(x) = x sin x, 0 x 4(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of f. f(x)=1+1x1x250E51E(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of f. f(x)=ex1ex53E54E55E(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of f. f(x) = earctan xSuppose the derivative of a function f is f(x) = (x + 1)2 (x 3)5 (x 6)4. On what interval is f increasing?58E59E60E61E62E63E64E65E66EThe president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its first and second derivatives.68ELet K(t) be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, K(8) K(7) or K(3) K(2)? Is the graph of K concave upward or concave downward? Why?70E71E72EFind a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of 3 at x = 2 and a local minimum value of 0 at x = 1.74E(a) If the function f(x) = x3 + ax2 + bx has the local minimum value 293 at x=1/3, what are the values of a and b? (b) Which of the tangent lines to the curve in part (a) has the smallest slope?76EShow that the curve y = (1 + x)/(1 + x2) has three points of inflection and they all lie on one straight line.Show that the curves y = ex and y = ex touch the curve y = ex sin x at its inflection points.79E80EAssume that all of the functions are twice differentiable and the second derivatives are never 0. (a) If f and g are positive, increasing, concave upward functions on I, show that the product function fg is concave upward on I. (b) Show that part (a) remains true if f and g are both decreasing. (c) Suppose f is increasing and g is decreasing. Show, by giving three examples, that fg may be concave upward, concave downward, or linear. Why doesnt the argument in parts (a) and (b) work in this case?82E83E(a) Show that ex 1 + x for x 0. (b) Deduce that ex1+x+12x2forx0. (c) Use mathematical induction to prove that for x 0 and any positive integer n, ex1+x+x22!++xnn!85EFor what values of c does the polynomial P(x) = x4 + cx3 + x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases?87E88E89E90E91E92EThe three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f, g, and h whose values at 0 are all 0 and, for x 0, f(x)=x4sin1xg(x)=x(2+sin1x)h(x)=x4(2+sin1x) (a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that f has neither a local maximum nor a local minimum at 0, g has a local minimum, and h has a local maximum.Given that limxaf(x)=0limxag(x)=0limxah(x)=1limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. (a) limxaf(x)g(x) (b) limxaf(x)p(x) (c) limxah(x)p(x) (d) limxap(x)f(x) (e) limxap(x)q(x)Given that limxaf(x)=0limxag(x)=0limxah(x)=1limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. (a) limxa[f(x)p(x)] (b) limxa[h(x)p(x)] (c) limxa[p(x)q(x)]3EGiven that limxaf(x)=0limxag(x)=0limxah(x)=1limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. (a) limxa[f(x)]g(x) (b) limxa[f(x)]p(x) (c) limxa[h(x)]p(x) (d) limxa[p(x)]f(x) (e) limxa[p(x)]q(x) (f) limxap(x)q(x)5E6EThe graph of a function f and its tangent line at 0 are shown. What is the value of limx0f(x)ex1?Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx3x3x29Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx4x22x8x4Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx2x3+8x+2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx1x32x2+1x3112EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx(/2)+cosx1sinxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0tan3xsin2x15E16E17EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim1+cos1cosFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxlnxx20EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0+lnxxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxlnxx2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limt1t81t5124EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx01+2x14xxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limueu/10u3Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0ex1xx2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0sinhxxx329E30E31EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx(lnx)2xFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0x3x3x1Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0cosmxcosnxx2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0ln(1+x)cosx+ex136E37E38E39E40EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0cosx1+12x2x442EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxxsin(/x)Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxxex/2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0sin5xcsc3x46EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxx3ex248EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx1+lnxtan(x/2)50EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx1(xx11lnx)Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0(cscxcotx)53E54E55EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx1+[ln(x71)ln(x51)]57E58E59E60E61EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxx(ln2)/(1+lnx)Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxx1/xFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxxexFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0+(4x+1)cotx66E67E68E69E70E71E72EProve that limxexxn= for any positive integer n. This shows that the exponential function approaches infinity faster than any power of x.74E75E76E77E78E79ELight enters the eye through the pupil and strikes the retina, where photoreceptor cells sense light and color. W. Stanley Stiles and B. H. Crawford studied the phenomenon in which measured brightness decreases as light enters farther from the center of the pupil. (See the figure.) A light beam A that enters through the center of the pupil measures brighter than a beam B entering near the edge of the pupil. They detailed their findings of this phenomenon, known as the StilesCrawford effect of the first kind, in an important paper published in 1933. In particular, they observed that the amount of luminance sensed was not proportional to the area of the pupil as they expected. The percentage P of the total luminance entering a pupil of radius r mm that is sensed at the retina can be described by P=110r2r2ln10 where is an experimentally determined constant, typically about 0.05. (a) What is the percentage of luminance sensed by a pupil of radius 3 mm? Use = 0.05. (b) Compute the percentage of luminance sensed by a pupil of radius 2 mm. Does it make sense that it is larger than the answer to part (a)? (c) Compute limx0+P. Is the result what you would expect? Is this result physically possible? Source: Adapted from W. Stiles and B. Crawford, The Luminous Efficiency of Rays Entering the Eye Pupil at Different Points. Proceedings of the Royal Society of London, Series B: Biological Sciences 112 (1933): 42850.81E82E83E84E85E86E87E88E89E90E91ELet f(x)={xxifx01ifx=0 (a) Show that f is continuous at 0. (b) Investigate graphically whether f is differentiable at 0 by zooming in several limes toward the point (0, 1) on the graph of f. (c) Show that f is not differentiable at 0. How can you reconcile this fact with the appearance of the graphs in part (b)?Use the guidelines of this section to sketch the curve. y = x3 + 3x2Use the guidelines of this section to sketch the curve. y = 2 + 3x2 x33E4E5E6E7EUse the guidelines of this section to sketch the curve. y = (4 x2)59E10EUse the guidelines of this section to sketch the curve. y=xx223x+x212E13EUse the guidelines of this section to sketch the curve. y=1x2415E16E17E18E19EUse the guidelines of this section to sketch the curve. y=x3x221E22E23E24E25E26EThe table gives the population of the world P(t), in millions, where t is measured in years and t = 0 corresponds to the year 1900. t Population (millions) 0 1650 10 1750 20 1860 30 2070 40 2300 50 2560 60 3040 70 3710 80 4450 90 5280 100 6080 110 6870 (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing device to find a cubic function (a third-degree polynomial) that models the data. (c) Use your model in part (b) to find a model for the rate of population growth. (d) Use part (c) to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (e) In Section 1.1 we modeled P(t) with the exponential function f(t)=(1.43653109)(1.01395)t Use this model to find a model for the rate of population growth. (f) Use your model in part (e) to estimate the rate of growth in 1920 and 1980. Compare with your estimates in parts (a) and (d). (g) Estimate the rate of growth in 1985.Use the guidelines of this section to sketch the curve. y=xx21Use the guidelines of this section to sketch the curve. y = x 3x1/3Use the guidelines of this section to sketch the curve. y = x5/3 5x2/331EUse the guidelines of this section to sketch the curve. y=x3+1333E34E35EUse the guidelines of this section to sketch the curve. y = 2x tan x, /2 x /237E38EUse the guidelines of this section to sketch the curve. y=sinx1+cosx40E41EUse the guidelines of this section to sketch the curve. y = (1 x)ex43E44E45E46E47EUse the guidelines of this section to sketch the curve. y = ex/x249E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67EUse the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y=x3(x+1)269EUse the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y = 1 x + e1+x/371E72E73E74E75E76E1EProduce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f and f to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = 2x6 + 5x5 + 140x3 110x23E4E5E6E7E8E9E10E11E12E13ESketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values. f(x)=(2x+3)2(x2)x3(x5)215E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E(a) Investigate the family of polynomials given by the equation f(x) = 2x3 + cx2 + 2x. For what values of c does the curve have maximum and minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the curve y = x x3. Illustrate by graphing this curve and several members of the family.Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the one at the right, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in your table, estimate the answer to the problem. (b) Use calculus to solve the problem and compare with your answer to part (a). First number Second number Product 1 22 22 2 21 42 3 20 60Find two numbers whose difference is 100 and whose product is a minimum.Find two positive numbers whose product is 100 and whose sum is a minimum.The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?What is the maximum vertical distance between the line y = x + 2 and the parabola y = x2 for 1 x 2?What is the minimum vertical distance between the parabolas y = x2 + 1 and y = x x2?Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible.9EThe rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P=100II2+I+4 where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).12E13EA box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.16E17EA farmer wants to fence in a rectangular plot of land adjacent to the north wall of his bam. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs 20 per linear foot to install and the farmer is not willing to spend more than 5000, find the dimensions for the plot that would enclose the most area.19E20E21EFind the point on the curve y=x that is closest to the point (3, 0).Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).24EFind the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.27E28E29E30EA right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.33EA Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.62.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions of the poster with the smallest area.A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum?Answer Exercise 37 if one piece is bent into a square and the other into a circle.39E40EA cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h=13H.An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with a plane, then the magnitude of the force is F=Wsin+cos where is a constant called the coefficient of friction. For what value of is F smallest?If a resistor of R ohms is connected across a battery of E volts with internal resistance r ohms, then the power (in watts) in the external resistor is P=E2R(R+r)2 If E and r are fixed but R varies, what is the maximum value of the power?For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u (u v), then the time required to swim a distance L is L/(v u) and the total energy E required to swim the distance is given by E(v)=av3Lvu where a is the proportionality constant. (a) Determine the value of v that minimizes E. (b) Sketch the graph of E. Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% greater than the current speed.